Chaotic fibers

In general, modes of a waveguide have "scars", which correspond to the classical trajectories. The scars may avoid the core, then the mode is not coupled, and it is vain to excite such a mode in the double-clad fiber amplifier. The scars can be distributed more or less uniformly in so-called chaotic fibers^[9] have more complicated cross-sectional shape and provide more uniform distribution of intensity in the inner cladding, allowing efficient use of the pump light. However, the scarring takes place even in chaotic fibers.

Spiral shape

Spiral-shaped cladding (blue), its chunk (red), and 3 segments of a ray (green).

Modes of spiral-shaped double-clad fiber.^[8]

An almost-circular shape with small spiral deformation seems to be the most efficient for chaotic fibers. In such a fiber, the angular momentum of a ray increases at each reflection from the smooth wall, until the ray hits the "chunk", at which the spiral curve is broken (see figure at right). The core, placed in vicinity of this chunk, is intercepted more regularly by all the rays compared to other chaotic fibers. This behavior of rays has an analogy in wave optics. In the language of modes, all the modes have non-zero derivative in vicinity of the chunk, and cannot avoid the core if it is placed there. One example of modes is shown in the figure below and to the right. Although some of modes show scarring and wide voids, none of these voids cover the core.

The property of DCFs with spiral-shaped cladding can be interpreted as conservation of angular momentum. The square of the derivative of a mode at the boundary can be interpreted as pressure. Modes (as well as rays) touching the spiral-shaped boundary transfer some angular momentum to it. This transfer of angular momentum should be compensated by pressure at the chunk. Therefore, no one mode can avoid the chunk. Modes can show strong scarring along the classical trajectories (rays) and wide voids, but at least one of scars should approach the chunk to compensate for the angular momentum transferred by the spiral part.

The interpretation in terms of angular momentum indicates the optimum size of the chunk. There is no reason to make the chunk larger than the core; a large chunk would not localize the scars sufficiently to provide coupling with the core. There is no reason to localize the scars within an angle smaller than the core: the small derivative to the radius makes the manufacturing less robust; the larger ${\displaystyle R'(\phi )}$ is, the larger the fluctuations of shape that are allowed without breaking the condition ${\displaystyle R'(\phi )>0}$. Therefore, the size of the chunk should be of the same order as the size of the core.

More rigorously, the property of the spiral-shaped domain follows from the theorem about boundary behavior of modes of the Dirichlet Laplacian.^[10] Although this theorem is formulated for the core-less domain, it prohibits the modes avoiding the core. A mode avoiding the core, then, should be similar to that of the core-less domain.

Stochastic optimization of the cladding shape confirms that an almost-circular spiral realizes the best coupling of pump into the core.^[11]

Tapered

Tapered double-clad fiber (T-DCF) is a fiber whose outer and inner claddings and core diameters vary smoothly with length. The core at narrow side of T-DCF supports propagation of the fundamental mode only, whereas at the wide side the core is able to guide many modes. However, it was shown experimentally that light launched into the narrow end of a T-DCF propagates into the wide core without any changes of mode content.^[12] As a result, at the wide (substantially multimode) end of T-DCF light propagates only in the lowest-order mode with excellent beam quality. Thus, tapered fiber is a unique and easy way to implement fundamental mode regime propagation (and amplification) in a multimode fiber.

Filling factor

Estimates of the pump efficiency in a double-clad fiber with ${\displaystyle F=0.8}$ (blue) and ${\displaystyle F=0.9}$ (red), discussed in^[2] compared to the results of the ray tracing simulations^[6](black curves).

The efficiency of absorption of pumping energy in the fiber is an important parameter of a double-clad fiber laser. In many cases this efficiency can be approximated with^[2]

${\displaystyle 1-\exp \left(-F{\frac {\pi r^{2}}{S}}\alpha L\right),}$

where

${\displaystyle S}$ is the cross-sectional area of the cladding

${\displaystyle r}$ is the radius of the core (which is taken to be circular)

${\displaystyle \alpha }$ is the absorption coefficient of pump light in the core

${\displaystyle L}$ is the length of the double-clad fiber, and

${\displaystyle F}$ is a dimensionless adjusting parameter, which is sometimes called the "filling factor"; ${\displaystyle 0<F<1}$.

The filling factor may depend on the initial distribution of the pump light, the shape of the cladding, and the position of the core within it.

The exponential behavior of the efficiency of absorption of pump in the core is not obvious. One could expect that some modes of the cladding (or some rays) are better coupled to the core than others; therefore, the "true" dependence could be a combination of several exponentials. Only comparison with simulations justifies this approximation, as shown in the figure above and to the right. In particular, this approximation does not work for circular fibers, see the initial work by Bedo et al., cited below. For chaotic fibers, ${\displaystyle F}$ approaches unity. The value of ${\displaystyle F}$ can be estimated by numerical analysis with propagation of waves, expansion by modes or by geometrical optics ray tracing, and values 0.8 and 0.9 are only empirical adjusting parameters, which provide good agreement of the simple estimate with numerical simulations for two specific classes of double-clad fibers: circular offset and rectangular. Obviously, the simple estimate above fails when the offset parameter becomes small compared to the size of cladding.

The filling factor ${\displaystyle F}$ approaches unity especially quickly in the spiral-shaped cladding, due to the special boundary behavior of the modes of the Dirichlet Laplacian.^[10] Designers of double-clad fiber look for a reasonable compromise between the optimized shape (for the efficient coupling of pump into the core) and the simplicity of the manufacturing of the preform used to draw the fibers.

The power scaling of a fiber laser is limited by unwanted nonlinear effects such as stimulated Brillouin scattering and stimulated Raman scattering. These effects are minimized when the fiber laser is short. For efficient operation, however, the pump should be absorbed in the core along the short length; the estimate above applies in this optimistic case. In particular, the higher the step in refractive index from inner to outer cladding, the better-confined the pump is. As a limiting case, the index step can be of order of two, from glass to air.^[13] The estimate with filling factor gives an estimate of how short an efficient double-clad fiber laser can be, due to reduction in size of the inner cladding.

Alternative structures

For good cladding shapes the filling factor ${\displaystyle F}$, defined above, approaches unity; the following enhancement is possible at various kinds of tapering of the cladding;^[14] non-conventional shapes of such cladding are suggested.^[15]

Planar waveguides with an active gain medium take an intermediate position between conventional solid-state lasers and double-clad fiber lasers. The planar waveguide may confine a multi-mode pump and a high-quality signal beam, allowing efficient coupling of the pump, and diffraction-limited output.^[7][16]